Finite Dimensional Deterministic Nonlinear Filters via Riccati Transformation and Volterra Series
M.T. Nihtilä · SIAM Journal on Control and Optimization · 1983
Filtering in two classes of nonlinear differential and difference systems is studied. The system structures admit the representation of the optimal recursive filter in the form of a finite dimensional differential or difference system. The filtering problem is posed as a set of fixed interval optimization problems. The deterministic least-squares problem statement results in the filter which is an ordinary nonstochastic differential or difference system driven by the observed signal. No stochastic concepts are used. The system classes considered are described by two linear subsystems with a polynomial link map between them. In one class the link map affects the input of the latter subsystem. In the other class the coefficient matrix of the latter subsystem is a polynomial in the state of the preceding subsystem satisfying a Lie algebraic nilpotency condition. The former class is a special case of the systems described by finite Volterra series for which finite dimensionality of the optimal filter is also proved. Some examples and comparisons on the basis of examples with the corresponding stochastic systems admitting finite dimensional optimal conditional mean filters studied recently are performed. The comparisons show that except in some special cases including a linear one the deterministic least-squares filter and the stochastic conditional mean filter are not equivalent.